Пры рашэнні задач часта бывае карысна ведаць дакладныя алгебраічныя выразы для значэнняў трыганаметрычных функцый, у першую чаргу для таго, каб прадставіць рашэнне праз радыкалы (карані), што адкрывае магчымасці для далейшага спрашчэння.
Усе значэнні сінусаў, косінусаў і тангенсаў вуглоў, кратных 3°, выражаюцца ў радыкалах. Гэтыя значэнні атрыманы шляхам прымянення тоеснасцей для палавіннага вугла, двайнога вугла, а таксама формул для сумы і рознасці вуглоў са значэннямі 0°, 30°, 36°, і 45°.
Заўвага: градусы і радыяны звязаны суадносінамі 1° = π/180 радыян.
Згодна з тэарэмай Нівена[1], адзінымі рацыянальнымі значэннямі функцыі сінуса пры рацыянальным аргуменце (у градусах) з’яўляюцца лікі 0, 1/2, і 1.
Значэнні сінуса, косінуса, тангенса, катангенса, секанса і касеканса для найбольш ужывальных вострых вуглоў прыведзены ў табліцы. («∞» азначае, што функцыя ў таком пункце не вызначана, а ў яго наваколлі імкнецца да бесканечнасці).
0°(0 рад) | 30° (π/6) | 45° (π/4) | 60° (π/3) | 90° (π/2) | 180° (π) | 270° (3π/2) | 360° (2π) | |
---|---|---|---|---|---|---|---|---|
sin
π 60
= cos
29
π
60
= sin
3
∘
= cos
87
∘
=
2
(
3
1 ) (
5
− 1 ) − 2 (
3
− 1 )
5 +
5
16
,
{\displaystyle \sin {\frac {\pi }{60}}=\cos {\frac {29,\pi }{60}}=\sin 3^{\circ }=\cos 87^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}-1)-2({\sqrt {3}}-1){\sqrt {5+{\sqrt {5}}}}}{16}},}
cos
π 60
= sin
29
π
60
= cos
3
∘
= sin
87
∘
=
2
(
3
− 1 ) (
5
− 1 ) + 2 (
3
1 )
5 +
5
16
,
{\displaystyle \cos {\frac {\pi }{60}}=\sin {\frac {29,\pi }{60}}=\cos 3^{\circ }=\sin 87^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}-1)+2({\sqrt {3}}+1){\sqrt {5+{\sqrt {5}}}}}{16}},}
tg
π 60
= ctg
29
π
60
= tg
3
∘
= ctg
87
∘
=
2 (
5
2 ) −
3
(
5
3 ) + ( 2 −
3
) (
3
(
5
1 ) − 2 )
5 − 2
5
2
,
{\displaystyle \operatorname {tg} {\frac {\pi }{60}}=\operatorname {ctg} {\frac {29,\pi }{60}}=\operatorname {tg} 3^{\circ }=\operatorname {ctg} 87^{\circ }={\frac {2({\sqrt {5}}+2)-{\sqrt {3}}({\sqrt {5}}+3)+(2-{\sqrt {3}})({\sqrt {3}}({\sqrt {5}}+1)-2){\sqrt {5-2{\sqrt {5}}}}}{2}},}
ctg
π 60
= tg
29
π
60
= ctg
3
∘
= tg
87
∘
=
2 ( 2 (
5
2 ) +
3
(
5
3 ) ) + (
3
(
5
− 1 ) + 2 )
2 ( 25 + 11
5
)
4
,
{\displaystyle \operatorname {ctg} {\frac {\pi }{60}}=\operatorname {tg} {\frac {29,\pi }{60}}=\operatorname {ctg} 3^{\circ }=\operatorname {tg} 87^{\circ }={\frac {2(2({\sqrt {5}}+2)+{\sqrt {3}}({\sqrt {5}}+3))+({\sqrt {3}}({\sqrt {5}}-1)+2){\sqrt {2(25+11{\sqrt {5}})}}}{4}},}
sin
π 30
= cos
7
π
15
= sin
6
∘
= cos
84
∘
=
6 ( 5 −
5
)
−
5
− 1
8
,
{\displaystyle \sin {\frac {\pi }{30}}=\cos {\frac {7,\pi }{15}}=\sin 6^{\circ }=\cos 84^{\circ }={\frac {{\sqrt {6(5-{\sqrt {5}})}}-{\sqrt {5}}-1}{8}},}
cos
π 30
= sin
7
π
15
= cos
6
∘
= sin
84
∘
=
2 ( 5 −
5
)
3
(
5
1 )
8
,
{\displaystyle \cos {\frac {\pi }{30}}=\sin {\frac {7,\pi }{15}}=\cos 6^{\circ }=\sin 84^{\circ }={\frac {{\sqrt {2(5-{\sqrt {5}})}}+{\sqrt {3}}({\sqrt {5}}+1)}{8}},}
tg
π 30
= ctg
7
π
15
= tg
6
∘
= ctg
84
∘
=
2 ( 5 −
5
)
−
3
(
5
− 1 )
2
,
{\displaystyle \operatorname {tg} {\frac {\pi }{30}}=\operatorname {ctg} {\frac {7,\pi }{15}}=\operatorname {tg} 6^{\circ }=\operatorname {ctg} 84^{\circ }={\frac {{\sqrt {2(5-{\sqrt {5}})}}-{\sqrt {3}}({\sqrt {5}}-1)}{2}},}
ctg
π 30
= tg
7
π
15
= ctg
6
∘
= tg
84
∘
=
2 ( 25 + 11
5
)
3
(
5
3 )
2
,
{\displaystyle \operatorname {ctg} {\frac {\pi }{30}}=\operatorname {tg} {\frac {7,\pi }{15}}=\operatorname {ctg} 6^{\circ }=\operatorname {tg} 84^{\circ }={\frac {{\sqrt {2(25+11{\sqrt {5}})}}+{\sqrt {3}}({\sqrt {5}}+3)}{2}},}
sin
π 20
= cos
9
π
20
= sin
9
∘
= cos
81
∘
=
2
(
5
1 ) − 2
5 −
5
8
,
{\displaystyle \sin {\frac {\pi }{20}}=\cos {\frac {9,\pi }{20}}=\sin 9^{\circ }=\cos 81^{\circ }={\frac {{\sqrt {2}}({\sqrt {5}}+1)-2{\sqrt {5-{\sqrt {5}}}}}{8}},}
cos
π 20
= sin
9
π
20
= cos
9
∘
= sin
81
∘
=
2
(
5
1 ) + 2
5 −
5
8
,
{\displaystyle \cos {\frac {\pi }{20}}=\sin {\frac {9,\pi }{20}}=\cos 9^{\circ }=\sin 81^{\circ }={\frac {{\sqrt {2}}({\sqrt {5}}+1)+2{\sqrt {5-{\sqrt {5}}}}}{8}},}
tg
π 20
= ctg
9
π
20
= tg
9
∘
= ctg
81
∘
=
5
1 −
5 + 2
5
,
{\displaystyle \operatorname {tg} {\frac {\pi }{20}}=\operatorname {ctg} {\frac {9,\pi }{20}}=\operatorname {tg} 9^{\circ }=\operatorname {ctg} 81^{\circ }={{\sqrt {5}}+1-{\sqrt {5+2{\sqrt {5}}}}},}
ctg
π 20
= tg
9
π
20
= ctg
9
∘
= tg
81
∘
=
5
1 +
5 + 2
5
,
{\displaystyle \operatorname {ctg} {\frac {\pi }{20}}=\operatorname {tg} {\frac {9,\pi }{20}}=\operatorname {ctg} 9^{\circ }=\operatorname {tg} 81^{\circ }={{\sqrt {5}}+1+{\sqrt {5+2{\sqrt {5}}}}},}
sin
π 15
= cos
13
π
30
= sin
12
∘
= cos
78
∘
=
2 ( 5 +
5
)
−
3
(
5
− 1 )
8
,
{\displaystyle \sin {\frac {\pi }{15}}=\cos {\frac {13,\pi }{30}}=\sin 12^{\circ }=\cos 78^{\circ }={\frac {{\sqrt {2(5+{\sqrt {5}})}}-{\sqrt {3}}({\sqrt {5}}-1)}{8}},}
cos
π 15
= sin
13
π
30
= cos
12
∘
= sin
78
∘
=
6 ( 5 +
5
)
5
− 1
8
,
{\displaystyle \cos {\frac {\pi }{15}}=\sin {\frac {13,\pi }{30}}=\cos 12^{\circ }=\sin 78^{\circ }={\frac {{\sqrt {6(5+{\sqrt {5}})}}+{\sqrt {5}}-1}{8}},}
tg
π 15
= ctg
13
π
30
= tg
12
∘
= ctg
78
∘
=
3
( 3 −
5
) −
2 ( 25 − 11
5
)
2
,
{\displaystyle \operatorname {tg} {\frac {\pi }{15}}=\operatorname {ctg} {\frac {13,\pi }{30}}=\operatorname {tg} 12^{\circ }=\operatorname {ctg} 78^{\circ }={\frac {{\sqrt {3}}(3-{\sqrt {5}})-{\sqrt {2(25-11{\sqrt {5}})}}}{2}},}
ctg
π 15
= tg
13
π
30
= ctg
12
∘
= tg
78
∘
=
3
(
5
1 ) +
2 ( 5 +
5
)
2
,
{\displaystyle \operatorname {ctg} {\frac {\pi }{15}}=\operatorname {tg} {\frac {13,\pi }{30}}=\operatorname {ctg} 12^{\circ }=\operatorname {tg} 78^{\circ }={\frac {{\sqrt {3}}({\sqrt {5}}+1)+{\sqrt {2(5+{\sqrt {5}})}}}{2}},}
sin
7
π
60
= cos
23
π
60
= sin
21
∘
= cos
69
∘
=
−
2
(
3
− 1 ) (
5
1 ) + 2 (
3
1 )
5 −
5
16
,
{\displaystyle \sin {\frac {7,\pi }{60}}=\cos {\frac {23,\pi }{60}}=\sin 21^{\circ }=\cos 69^{\circ }={\frac {-{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}+1)+2({\sqrt {3}}+1){\sqrt {5-{\sqrt {5}}}}}{16}},}
cos
7
π
60
= sin
23
π
60
= cos
21
∘
= sin
69
∘
=
2
(
3
1 ) (
5
1 ) + 2 (
3
− 1 )
5 −
5
16
,
{\displaystyle \cos {\frac {7,\pi }{60}}=\sin {\frac {23,\pi }{60}}=\cos 21^{\circ }=\sin 69^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}+1)+2({\sqrt {3}}-1){\sqrt {5-{\sqrt {5}}}}}{16}},}
tg
7
π
60
= ctg
23
π
60
= tg
21
∘
= ctg
69
∘
=
2 ( 2 (
5
− 2 ) −
3
( 3 −
5
) ) + (
3
(
5
1 ) − 2 )
2 ( 25 − 11
5
)
4
,
{\displaystyle \operatorname {tg} {\frac {7,\pi }{60}}=\operatorname {ctg} {\frac {23,\pi }{60}}=\operatorname {tg} 21^{\circ }=\operatorname {ctg} 69^{\circ }={\frac {2(2({\sqrt {5}}-2)-{\sqrt {3}}(3-{\sqrt {5}}))+({\sqrt {3}}({\sqrt {5}}+1)-2){\sqrt {2(25-11{\sqrt {5}})}}}{4}},}
ctg
7
π
60
= tg
23
π
60
= ctg
21
∘
= tg
69
∘
=
2 ( 2 (
5
− 2 ) +
3
( 3 −
5
) ) + (
3
(
5
1 ) + 2 )
2 ( 25 − 11
5
)
4
,
{\displaystyle \operatorname {ctg} {\frac {7,\pi }{60}}=\operatorname {tg} {\frac {23,\pi }{60}}=\operatorname {ctg} 21^{\circ }=\operatorname {tg} 69^{\circ }={\frac {2(2({\sqrt {5}}-2)+{\sqrt {3}}(3-{\sqrt {5}}))+({\sqrt {3}}({\sqrt {5}}+1)+2){\sqrt {2(25-11{\sqrt {5}})}}}{4}},}
sin
2
π
15
= cos
11
π
30
= sin
24
∘
= cos
66
∘
=
3
(
5
1 ) −
2 ( 5 −
5
)
8
,
{\displaystyle \sin {\frac {2,\pi }{15}}=\cos {\frac {11,\pi }{30}}=\sin 24^{\circ }=\cos 66^{\circ }={\frac {{\sqrt {3}}({\sqrt {5}}+1)-{\sqrt {2(5-{\sqrt {5}})}}}{8}},}
cos
2
π
15
= sin
11
π
30
= cos
24
∘
= sin
66
∘
=
5
1 +
6 ( 5 −
5
)
8
,
{\displaystyle \cos {\frac {2,\pi }{15}}=\sin {\frac {11,\pi }{30}}=\cos 24^{\circ }=\sin 66^{\circ }={\frac {{\sqrt {5}}+1+{\sqrt {6(5-{\sqrt {5}})}}}{8}},}
tg
2
π
15
= ctg
11
π
30
= tg
24
∘
= ctg
66
∘
=
−
3
( 3 +
5
) +
2 ( 25 + 11
5
)
2
,
{\displaystyle \operatorname {tg} {\frac {2,\pi }{15}}=\operatorname {ctg} {\frac {11,\pi }{30}}=\operatorname {tg} 24^{\circ }=\operatorname {ctg} 66^{\circ }={\frac {-{\sqrt {3}}(3+{\sqrt {5}})+{\sqrt {2(25+11{\sqrt {5}})}}}{2}},}
ctg
2
π
15
= tg
11
π
30
= ctg
24
∘
= tg
66
∘
=
3
(
5
− 1 ) +
2 ( 5 −
5
)
2
,
{\displaystyle \operatorname {ctg} {\frac {2,\pi }{15}}=\operatorname {tg} {\frac {11,\pi }{30}}=\operatorname {ctg} 24^{\circ }=\operatorname {tg} 66^{\circ }={\frac {{\sqrt {3}}({\sqrt {5}}-1)+{\sqrt {2(5-{\sqrt {5}})}}}{2}},}
sin
3
π
20
= cos
7
π
20
= sin
27
∘
= cos
63
∘
=
−
2
(
5
− 1 ) + 2
5 +
5
8
,
{\displaystyle \sin {\frac {3,\pi }{20}}=\cos {\frac {7,\pi }{20}}=\sin 27^{\circ }=\cos 63^{\circ }={\frac {-{\sqrt {2}}({\sqrt {5}}-1)+2{\sqrt {5+{\sqrt {5}}}}}{8}},}
cos
3
π
20
= sin
7
π
20
= cos
27
∘
= sin
63
∘
=
2
(
5
− 1 ) + 2
5 +
5
8
,
{\displaystyle \cos {\frac {3,\pi }{20}}=\sin {\frac {7,\pi }{20}}=\cos 27^{\circ }=\sin 63^{\circ }={\frac {{\sqrt {2}}({\sqrt {5}}-1)+2{\sqrt {5+{\sqrt {5}}}}}{8}},}
tg
3
π
20
= ctg
7
π
20
= tg
27
∘
= ctg
63
∘
=
5
− 1 −
5 − 2
5
,
{\displaystyle \operatorname {tg} {\frac {3,\pi }{20}}=\operatorname {ctg} {\frac {7,\pi }{20}}=\operatorname {tg} 27^{\circ }=\operatorname {ctg} 63^{\circ }={{\sqrt {5}}-1-{\sqrt {5-2{\sqrt {5}}}}},}
ctg
3
π
20
= tg
7
π
20
= ctg
27
∘
= tg
63
∘
=
5
− 1 +
5 − 2
5
,
{\displaystyle \operatorname {ctg} {\frac {3,\pi }{20}}=\operatorname {tg} {\frac {7,\pi }{20}}=\operatorname {ctg} 27^{\circ }=\operatorname {tg} 63^{\circ }={{\sqrt {5}}-1+{\sqrt {5-2{\sqrt {5}}}}},}
sin
11
π
60
= cos
19
π
60
= sin
33
∘
= cos
57
∘
=
2
(
3
1 ) (
5
− 1 ) + 2 (
3
− 1 )
5 +
5
16
,
{\displaystyle \sin {\frac {11,\pi }{60}}=\cos {\frac {19,\pi }{60}}=\sin 33^{\circ }=\cos 57^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}-1)+2({\sqrt {3}}-1){\sqrt {5+{\sqrt {5}}}}}{16}},}
cos
11
π
60
= sin
19
π
60
= cos
33
∘
= sin
57
∘
=
−
2
(
3
− 1 ) (
5
− 1 ) + 2 (
3
1 )
5 +
5
16
,
{\displaystyle \cos {\frac {11,\pi }{60}}=\sin {\frac {19,\pi }{60}}=\cos 33^{\circ }=\sin 57^{\circ }={\frac {-{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}-1)+2({\sqrt {3}}+1){\sqrt {5+{\sqrt {5}}}}}{16}},}
tg
11
π
60
= ctg
19
π
60
= tg
33
∘
= ctg
57
∘
=
− 2 (
5
2 ) +
3
( 3 +
5
) + ( 2 −
3
) (
3
(
5
1 ) − 2 )
5 − 2
5
2
,
{\displaystyle \operatorname {tg} {\frac {11,\pi }{60}}=\operatorname {ctg} {\frac {19,\pi }{60}}=\operatorname {tg} 33^{\circ }=\operatorname {ctg} 57^{\circ }={\frac {-2({\sqrt {5}}+2)+{\sqrt {3}}(3+{\sqrt {5}})+(2-{\sqrt {3}})({\sqrt {3}}({\sqrt {5}}+1)-2){\sqrt {5-2{\sqrt {5}}}}}{2}},}
ctg
11
π
60
= tg
19
π
60
= ctg
33
∘
= tg
57
∘
=
− 2 ( 2 (
5
2 ) +
3
( 3 +
5
) ) + (
3
(
5
− 1 ) + 2 )
2 ( 25 + 11
5
)
4
,
{\displaystyle \operatorname {ctg} {\frac {11,\pi }{60}}=\operatorname {tg} {\frac {19,\pi }{60}}=\operatorname {ctg} 33^{\circ }=\operatorname {tg} 57^{\circ }={\frac {-2(2({\sqrt {5}}+2)+{\sqrt {3}}(3+{\sqrt {5}}))+({\sqrt {3}}({\sqrt {5}}-1)+2){\sqrt {2(25+11{\sqrt {5}})}}}{4}},}
sin
13
π
60
= cos
17
π
60
= sin
39
∘
= cos
51
∘
=
2
(
3
1 ) (
5
1 ) − 2 (
3
− 1 )
5 −
5
16
,
{\displaystyle \sin {\frac {13,\pi }{60}}=\cos {\frac {17,\pi }{60}}=\sin 39^{\circ }=\cos 51^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}+1)-2({\sqrt {3}}-1){\sqrt {5-{\sqrt {5}}}}}{16}},}
cos
13
π
60
= sin
17
π
60
= cos
39
∘
= sin
51
∘
=
2
(
3
− 1 ) (
5
1 ) + 2 (
3
1 )
5 −
5
16
,
{\displaystyle \cos {\frac {13,\pi }{60}}=\sin {\frac {17,\pi }{60}}=\cos 39^{\circ }=\sin 51^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}+1)+2({\sqrt {3}}+1){\sqrt {5-{\sqrt {5}}}}}{16}},}
tg
13
π
60
= ctg
17
π
60
= tg
39
∘
= ctg
51
∘
=
− 2 ( 2 (
5
− 2 ) +
3
( 3 −
5
) ) + (
3
(
5
1 ) + 2 )
2 ( 25 − 11
5
)
4
,
{\displaystyle \operatorname {tg} {\frac {13,\pi }{60}}=\operatorname {ctg} {\frac {17,\pi }{60}}=\operatorname {tg} 39^{\circ }=\operatorname {ctg} 51^{\circ }={\frac {-2(2({\sqrt {5}}-2)+{\sqrt {3}}(3-{\sqrt {5}}))+({\sqrt {3}}({\sqrt {5}}+1)+2){\sqrt {2(25-11{\sqrt {5}})}}}{4}},}
ctg
13
π
60
= tg
17
π
60
= ctg
39
∘
= tg
51
∘
=
− 2 ( 2 (
5
− 2 ) −
3
( 3 −
5
) ) + (
3
(
5
1 ) − 2 )
2 ( 25 − 11
5
)
4
,
{\displaystyle \operatorname {ctg} {\frac {13,\pi }{60}}=\operatorname {tg} {\frac {17,\pi }{60}}=\operatorname {ctg} 39^{\circ }=\operatorname {tg} 51^{\circ }={\frac {-2(2({\sqrt {5}}-2)-{\sqrt {3}}(3-{\sqrt {5}}))+({\sqrt {3}}({\sqrt {5}}+1)-2){\sqrt {2(25-11{\sqrt {5}})}}}{4}},}
sin
7
π
30
= cos
8
π
30
= sin
42
∘
= cos
48
∘
=
− (
5
− 1 ) +
6 ( 5 +
5
)
8
,
{\displaystyle \sin {\frac {7,\pi }{30}}=\cos {\frac {8,\pi }{30}}=\sin 42^{\circ }=\cos 48^{\circ }={\frac {-({\sqrt {5}}-1)+{\sqrt {6(5+{\sqrt {5}})}}}{8}},}
cos
7
π
30
= sin
8
π
30
= cos
42
∘
= sin
48
∘
=
3
(
5
− 1 ) +
2 ( 5 +
5
)
8
,
{\displaystyle \cos {\frac {7,\pi }{30}}=\sin {\frac {8,\pi }{30}}=\cos 42^{\circ }=\sin 48^{\circ }={\frac {{\sqrt {3}}({\sqrt {5}}-1)+{\sqrt {2(5+{\sqrt {5}})}}}{8}},}
tg
7
π
30
= ctg
8
π
30
= tg
42
∘
= ctg
48
∘
=
3
(
5
1 ) −
2 ( 5 +
5
)
2
,
{\displaystyle \operatorname {tg} {\frac {7,\pi }{30}}=\operatorname {ctg} {\frac {8,\pi }{30}}=\operatorname {tg} 42^{\circ }=\operatorname {ctg} 48^{\circ }={\frac {{\sqrt {3}}({\sqrt {5}}+1)-{\sqrt {2(5+{\sqrt {5}})}}}{2}},}
ctg
7
π
30
= tg
8
π
30
= ctg
42
∘
= tg
48
∘
=
3
( 3 −
5
) +
2 ( 25 − 11
5
)
2
,
{\displaystyle \operatorname {ctg} {\frac {7,\pi }{30}}=\operatorname {tg} {\frac {8,\pi }{30}}=\operatorname {ctg} 42^{\circ }=\operatorname {tg} 48^{\circ }={\frac {{\sqrt {3}}(3-{\sqrt {5}})+{\sqrt {2(25-11{\sqrt {5}})}}}{2}},}
tg
π 120
= ctg
59
π
120
= tg
1.5
∘
= ctg
88.5
∘
=
8 −
2 ( 2 −
3
) ( 3 −
5
)
−
2 ( 2 +
3
) ( 5 +
5
)
8 +
2 ( 2 −
3
) ( 3 −
5
)
2 ( 2 +
3
) ( 5 +
5
)
,
{\displaystyle \operatorname {tg} {\frac {\pi }{120}}=\operatorname {ctg} {\frac {59,\pi }{120}}=\operatorname {tg} 1.5^{\circ }=\operatorname {ctg} 88.5^{\circ }={\sqrt {\frac {8-{\sqrt {2(2-{\sqrt {3}})(3-{\sqrt {5}})}}-{\sqrt {2(2+{\sqrt {3}})(5+{\sqrt {5}})}}}{8+{\sqrt {2(2-{\sqrt {3}})(3-{\sqrt {5}})}}+{\sqrt {2(2+{\sqrt {3}})(5+{\sqrt {5}})}}}}},}
cos
π 240
= sin
119
π
240
= cos
0.75
∘
= sin
89.25
∘
=
1 16
(
2 −
2 +
2
(
2 ( 5 +
5
)
3
( 1 −
5
)
)
{\displaystyle \cos {\frac {\pi }{240}}=\sin {\frac {119,\pi }{240}}=\cos 0.75^{\circ }=\sin 89.25^{\circ }={\frac {1}{16}}\left({\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}\left({\sqrt {2(5+{\sqrt {5}})}}+{\sqrt {3}}(1-{\sqrt {5}})\right)+\right.}
2 +
2 +
2
(
6 ( 5 +
5
)
5
− 1
)
)
,
{\displaystyle \left.+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}\left({\sqrt {6(5+{\sqrt {5}})}}+{\sqrt {5}}-1\right)\right),}
cos
π 17
= sin
15
π
34
=
1 8
2
(
2
3
17
−
2 ( 85 + 19
17
)
17
2 ( 17 −
17
)
17
15
)
.
{\displaystyle \cos {\frac {\pi }{17}}=\sin {\frac {15,\pi }{34}}={\frac {1}{8}}{\sqrt {2\left(2{\sqrt {3{\sqrt {17}}-{\sqrt {2(85+19{\sqrt {17}})}}+17}}+{\sqrt {2(17-{\sqrt {17}})}}+{\sqrt {17}}+15\right)}}.}